| L(s) = 1 | + (0.984 + 0.173i)2-s + (1.08 − 1.34i)3-s + (0.939 + 0.342i)4-s + (−3.12 − 1.13i)5-s + (1.30 − 1.13i)6-s + (1.24 − 2.33i)7-s + (0.866 + 0.5i)8-s + (−0.629 − 2.93i)9-s + (−2.88 − 1.66i)10-s + (−0.206 − 0.567i)11-s + (1.48 − 0.893i)12-s + (0.294 − 0.807i)13-s + (1.63 − 2.07i)14-s + (−4.94 + 2.97i)15-s + (0.766 + 0.642i)16-s + (−3.20 + 5.55i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.628 − 0.777i)3-s + (0.469 + 0.171i)4-s + (−1.39 − 0.509i)5-s + (0.533 − 0.464i)6-s + (0.472 − 0.881i)7-s + (0.306 + 0.176i)8-s + (−0.209 − 0.977i)9-s + (−0.911 − 0.526i)10-s + (−0.0622 − 0.171i)11-s + (0.428 − 0.257i)12-s + (0.0815 − 0.224i)13-s + (0.437 − 0.555i)14-s + (−1.27 + 0.768i)15-s + (0.191 + 0.160i)16-s + (−0.778 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60216 - 1.24314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60216 - 1.24314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (-1.08 + 1.34i)T \) |
| 7 | \( 1 + (-1.24 + 2.33i)T \) |
| good | 5 | \( 1 + (3.12 + 1.13i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (0.206 + 0.567i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.294 + 0.807i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.20 - 5.55i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.68 + 2.70i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.85 + 1.20i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.29 - 9.06i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.570 + 1.56i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 0.200T + 37T^{2} \) |
| 41 | \( 1 + (9.49 + 3.45i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.00 - 11.3i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.82 + 3.21i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.60 + 1.50i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.24 - 1.88i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.28 - 6.27i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.879 - 4.99i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.30 + 2.48i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.26iT - 73T^{2} \) |
| 79 | \( 1 + (-1.06 + 6.03i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (6.72 - 2.44i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.03 - 1.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.56 - 1.50i)T + (91.1 + 33.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.41160052584837214790605556351, −10.62769947958632405267618629548, −8.857701083608587181233305687759, −8.221282569177862516158719370685, −7.35890499272192478608242301003, −6.72408892195605616586183032997, −5.05523286431831050443587075176, −4.03311143639384582781583802202, −3.13317487134509999680621603726, −1.15547295632594753128272760183,
2.53982454350865163902586101758, 3.44159074628606759677864831597, 4.52244306815744655218129412064, 5.31507636763542897698189661323, 6.97020237634196338530199057406, 7.81902599314591956983391161373, 8.747110274155438698759378595248, 9.757023729715983525800188834686, 10.96034327510464507896472856998, 11.58374315483759900966759078304
